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FOURIER CIRCUIT ANALYSIS,
FILTERS AND ATTENUATORS

• Any periodic function f (t) can be expressed into an infinite trigonometric
series if it satisfies the following conditions:
• 1. It is well defined and single valued, except possibly at a finite number of
points.
• 2. It has a finite number of discontinuities in the period T.
• 3. It has a finite number of positive and negative maxima in the period T.
• These conditions are known as Dirichlet’s conditions.
• Any function f (t) which is periodic and satisfies the Dirichlet’s condition,
can be represented in terms of sine and cosine functions.

• Complex Fourier Spectrum :
A periodic function with period T has frequency components of angular
frequencies w, 2w, 3w,… nw where ω =2π/T Thus, the periodic function f (t) possesses
its spectrum of frequencies, known as line spectrum. The spectrum exists only at w,
2w, 3w,… etc. Thus, the spectrum is not a continuous curve but exists only at some
discrete values of w. The line spectrum is the plot which shows the variation of
magnitude and phase of the function w.r.t n
(i) Magnitude Spectrum: It is a plot of magnitude cn versus nw where
(ii) Phase Spectrum: It is a plot of phase angle φn versus nw
Where

• WAVEFORM SYMMETRY
There are four types of waveform symmetry in any function:
(i) Even symmetry (ii) Odd symmetry (iii) Half-wave symmetry
(iv) Quarter-wave symmetry
• Even Symmetry:
A function is said to have even symmetry if (f)t = f(-t) as shown.Even
functions are symmetrical about the vertical axis.
• when f(t) is an even function,

• Odd Symmetry:
A function is said to have odd symmetry if f (t) = f (-t) as shown. Odd
functions are symmetrical about the origin.
When f (t) is an odd function,
• Half-Wave Symmetry:
A function is said to have half-wave symmetry if as shown. The
values of f (t) are equal and opposite during the interval 0<t<(T/2) and (T/2) <t<T
Similarly, the values of f (t) are also equal and opposite during the interval
(-T/2)<t<0 and (-T/2)<t<T.

• When f (t) has half-wave symmetry,
if n is even
if n is odd
if n is odd
• Quarter-Wave Symmetry:
A function is said to have quarter-wave symmetry if it is a combination of
even, odd and half-wave symmetry as shown

• When f (t) has quarter-wave symmetry (with combination of even and half-
wave symmetry)
• When f (t) has quarter-wave symmetry (with combination of odd and half-
wave symmetry)

• EXPONENTIAL FOURIER SERIES

• AVERAGE VALUE OF A PERIODIC COMPLEX WAVE
• RMS VALUE OF PERIODIC COMPLEX WAVE

• where V1,V2………… Vn are the rms values of the harmonic components of
the wave. Hence, the rms value of any periodic complex wave is the square
root of sum of squares of rms values of each harmonic component and the
square of constant terms.

• POWER SUPPLIED BY COMPLEX WAVE

• CLASSIFICATION OF FILTERS
• On the basis of frequency characteristics, filters are classified into four
categories: (i) Low-pass filter (ii) High-pass filter (iii) Band-pass filter (iv) Band-
stop filter
• A low-pass filter allows all frequencies up to a certain cut-off frequency to
pass through it and attenuates all the other frequencies above the cut-off
frequency.
• A high-pass filter attenuates all the frequency below the cut-off frequency
and allows all other frequencies above the cut-off frequency to pass through
it.
• A band-pass filter allows a limited band of frequencies to pass through it and
attenuates all other frequencies below or above the frequency band.
• A band-stop filter attenuates a limited band of frequencies but allows all
other frequencies to pass through it.

• T-NETWORK
• Characteristic Impedance:
For a T network, the value of input impedance when it
is terminated by characteristic impedance Z0, is given by
Characteristic impedance can also be expressed in terms of open-circuit
impedance Zoc and short-circuit impedance Zsc.
Open-circuit impedance:
Short-circuit impedance:

• Propagation Constant:
The propagation constant γ( )of the network in is
given by
The propagation constant of a symmetrical p network is same as that of a
symmetrical T network.

• -
For a pie network, the value of input impedance
when it is terminated by impedance Z0, is given by
• Relation between Z0T and Z0π
• Characteristic impedance can also be expressed in terms of open-circuit
impedance Zoc and short-circuit impedance Zsc.
• Open-circuit impedance

• Short-circuit impedance :
(since )
• Propagation Constant:
The propagation constant of a symmetrical p network is same as that of
a symmetrical T network.

CONSTANT-k LOW PASS FILTER
• A T or pie network is said to be of the constant k type if Z1 and Z2 are
opposite types of reactances satisfying the relation
• k is often referred to as design impedance or nominal impedance of the
constant-k filter. The constant-k, T or p-type filter is also known as the
prototype filter because other complex networks can be derived from it.
• In constant-k low pass filter,

• Nominal Impedance(k):
• Cut-off Frequency:
The cut-off frequencies are obtained when
Hence, the pass band starts at f = 0 and continues up to the cut-off frequency
fc . All the frequencies above f c are in the attenuation or stop band.

• Attenuation Constant:
• Phase Constant:

• Design of Filter:

CONSTANT-k HIGH-PASS FILTER
A constant-k high-pass filter is obtained by changing the positions of series
and shunt arm of the constant-k low-pass filter, shows a constant-k, T and pie
section, high-pass filter.
In a constant-k high-pass filter
Nominal Impedence(k):

• Cut-off Frequency:
• Hence, the filter passes all the frequencies beyond f c . The pass band starts
at f = f c and continues up to infinite frequency. All the frequencies below the
cut-off frequency lie in the attenuation or stop band

• BAND-PASS FILTER
A band-pass filter attenuates all the frequencies below a lower cutoff
frequency and above an upper cut-off frequency. It passes a band of
frequencies without attenuation. A band pass filter is obtained by using a low
pass filter followed by a high-pass filter.

• BAND-STOP FILTER:
A band-stop filter attenuates a specified band of frequencies and allows all
frequencies below and above this band. A band-stop filter is realised by
connecting a low-pass filter in parallel with a high-pass filter.

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